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Voltmeter in Series? Voltmeter in series? Zoltan Gingl and Robert Mingesz Department of Technical Informatics, University of Szeged, Árpád tér 2, 6720 Szeged, Hungary E-mail: [email protected] Abstract A recent physics challenge shows a circuit, where a voltmeter is connected in series. Indeed, real voltmeters have finite input resistance, therefore one may think that they can be used as resistors. In addition, voltmeters measure the voltage difference between their terminals, therefore it seems to be possible to calculate the current flowing through them. Is it okay? Does it make the voltmeter more universal? Are there any hidden secrets? How it is related to high-quality physics and STEM education, which are increasingly important in the modern world? Doesn’t it approve an improper use that one can never see in any textbook and application? Suggesting and teaching such uncommon solutions doesn’t generate undesired attitude? On the other hand, can it make the development of creativity and understanding harder if the students are taught to follow always the application rules? We do think it bears some discussion. The voltmeter-in-series circuit Figure 1 shows the circuit of the challenge “Why so series?” [1, 2]. R 3R V 0.5 V 3R R V A 3 V 6 mA Figure 1. Circuit of the challenge [1, 2]. The meters show 6 mA, 0.5 V and 3.0 V. Looking at the voltmeter in series can be confusing, since it is used as a conducting element. Although real voltmeters have finite input resistance and some current can flow through them, it is taught, that voltmeters should always be connected in parallel to the component on which the voltage drop is to be measured regardless of the value of their input resistance. Following the anomalous use of the voltmeter shown above, one could connect a non-ideal ammeter in parallel and one could come up with a circuit, where two different voltage sources with non-zero internal resistances are connected in parallel, see figure 2. Looking at these circuits likely makes most of us uncomfortable, since we know that connecting voltage generators and ammeters in parallel can even cause damage. ? ? V R A R V1 V2 Figure 2. Theoretically it seems to be allowed to connect non-ideal ammeters and voltage sources with certain non-zero internal resistances in parallel. Other examples can also be mentioned. Real capacitors can have some DC conductance (typically modelled by a resistor in parallel) and inductors can have considerable series resistance [3]. So, in theory, they could be used as resistors in DC circuits just like the voltmeter in figure 1. However, it is never done in practice. We believe, it is worth to discuss the methodology behind the voltmeter-in-series example and especially its relation to education. Ideal and real voltmeters and ammeters It is well known that a perfect instrument must not affect the operation of the observed system. Therefore, an ideal voltmeter behaves like an open circuit, while an ideal ammeter acts as a short circuit. In electronics, real components are modelled by combinations of ideal components. A real voltmeter is represented by an ideal voltmeter and a resistor, or more generally, an impedance, connected in parallel (see figure 3). So, it acts like an impedance, it can conduct current. Similar is true for real ammeters; they also function as impedances. From this point of view, only their very different practical impedance values separate them. ZIN,A ZIN,V V ZIN,V ZIN,A A Figure 3. A real voltmeter or ammeter can be modelled as a parallel or serial combination of an ideal meter and an impedance, respectively. Both function as impedances in the circuit, but their real values are very different. Using voltmeters In practice the impact of the input impedance of voltmeters on the system operation must be negligible. Therefore, there is always a certain limit of application, since the required accuracy must not be compromised by the presence of the input impedance. For most digital multimeters (DMM) its value is 10 MΩ, while oscilloscope inputs typically load the signal source with 1 MΩ. Note, that in the challenge the input resistance of the voltmeters equals 500 Ω, which is 20000 times less than the typical value for today’s voltmeters. In some exceptional cases low input impedance can be desirable. High speed signals may need proper termination to avoid reflections [4], therefore oscilloscope inputs can have a very low input impedance of 50 Ω. Low impedance (Lo Z) mode of voltmeters also exist to cancel the so-called ghost voltages typically present as a parasitic voltage in non-energized mains circuits [5]. Although the input impedance can be low, the voltmeters are always connected in parallel. Can the input impedance of a voltmeter be taken into account? One can think that effect of the input impedance ZIN of the voltmeter can be taken into account and the voltmeter can even be used to measure the current flowing through it. For instance, the VG voltage of a generator connected to the voltmeter via a series resistor R in figure 4 can be calculated as 푅 + 푍퐼푁 푉퐺 = 푉 (1) 푍퐼푁 where V is the measured voltage. R VG ZIN V Figure 4. If the series resistor R and the input impedance ZIN is known, VG can be determined using the value measured by the voltmeter. Similarly, if a current I flows through the voltmeter’s input impedance (figure 5), its value can be expressed as 푉 퐼 = (2) 푍퐼푁 I ZIN V Figure 5. The voltmeter shows the voltage equal to the current multiplied by the input impedance. For example, consider a 4.5 digit DMM having 10 MΩ input impedance according to the datasheet. If it displays a value of 1.0000 V and R is 10 MΩ, then VG is calculated as 2.0000 V. In the same case, the current flowing through ZIN equals 0.1000 μA. It seems to be very straightforward. However, doing so is not a good practice at all! Let us see why. Input impedance variants and specifications All instruments have accuracy specifications and defined normal operating conditions, otherwise the measured value would be unreliable and useless. Although the voltage measurement accuracy of a voltmeter is always given, in contrast, the input impedance tolerance is rarely specified. Therefore, using the nominal value in calculations introduces unknown errors and can even cause the loss of reliability. Let us see some examples. A leading manufacturer specifies the input impedance of their DMM only as >10 MΩ in DC voltage measurement mode (see the “Fluke 114, 115, 116 and 117 Digital Multimeters Extended specifications”, https://dam-assets.fluke.com/s3fs- public/2793260_6116_ENG_A_W.PDF ). We think the message is clear: do not assume anything about its actual accuracy, use it only, when its effect is negligible. In other words, don’t try to use the nominal value of the input impedance to compensate its effect or to calculate the current flowing through it. Note, that during AC voltage measurements the input impedance of the same instrument is given as >5 MΩ. The manual of our UT60H 4.5 digit DMM used in education says that the voltage measurement accuracy is 0.1%+5 counts (means 0.15% overall error for 1.0000 V), while the input impedance is specified as “Approx. 10 MΩ”. Some other DMMs’ datasheet may say “about 10 MΩ”, or “10 MΩ nominal” or just “10 MΩ”. We have measured the input impedance of this DMM as a function of the input voltage, see figure 6. 12 ] Ω 11 10 Input impedanceInput[M 9 0 10 20 30 40 50 60 Input voltage [V] Figure 6. Input resistance as a function of the input voltage for a 4.5 digit multimeter under test (UT60H). The value is different in the measurement ranges of 4 V, 40 V and 400 V. One can see that the value can be more than 10 % apart from the nominal value and it depends on the selected measurement range, too. So, using the measured 11.12 MΩ value in equation (1) and (2) gives 1.8993 V and 0.0899 μA in the abovementioned example instead of 2.0000 V and 0.1000 μA. Therefore, using the nominal value introduces an error of about 33 and 67 times higher than the error caused by the DMM itself! What is the reason for such surprising and significant variations in the input impedance that can make even electronics enthusiasts and professionals sometimes confused? There are typically two solutions in DMMs to set the input voltage range. Manual range selection is done by using a rotary switch that changes the voltage division ratio, see the left hand side circuit of figure 7. In this case the input voltage source always sees a constant load of close to 10 MΩ regardless of the range. Auto-ranging DMMs change the division ratio using integrated circuit switches, accordingly they typically employ a more suitable solution that can be seen on the right hand side of figure 7. The input impedance depends on the selected range in this case as we have observed. There are even such auto-ranging DMMs, for which the input impedance at the lowest ranges can be well above 1000 MΩ (all switches are off), while in other ranges their nominal impedance is still close to 10 MΩ. 10M VIN 10 10 10 9M 900k 90k 10k M M M VIN 9 99 999 V V VADC ADC V Figure 7. Principles of voltage divider circuits used in manual (left hand side) and auto- ranging voltmeters (right hand side) to select the division ratio of the input voltage VIN to provide a voltage VADC for analogue-to-digital conversion.
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